Quantum error correction and holographic information from bilocal holography

Abstract

Bilocal holography is a constructive approach to the higher spin theory holographically dual to O(N ) vector models. In contrast to other approaches to bulk reconstruction, bilocal holography does not take input from the dual gravitational theory. The resulting map is a complete bulk/boundary mapping in that it maps the complete set of O(N ) invariant degrees of freedom in the CFT, to the complete set of higher spin degrees of freedom. After restricting to a suitable code subspace we demonstrate that bilocal holography naturally reproduces the quantum error correcting properties of holography and it gives a robust bulk (entanglement wedge) reconstruction. A gauge invariant entangled pair of CFT degrees of freedom are naturally smeared over a semicircle in the bulk spacetime, which is highly suggestive of bit threads. Finally, we argue that finite N relations in the CFT, when interpreted in the dual AdS spacetime, can provide relations between degrees of freedom located near the boundary and degrees of freedom deep in the bulk.

Holography of information in AdS/CFT

“…(2.10) so that θ appearing in figure 1 is the same coordinate θ appearing in the map [39]. Consequently, varying p + 1 and p + 2 moves us along the semi-circle.…”

“…Finally, it was pointed out in [39] there is some freedom in the reconstruction of the bulk fields. This freedom will be used in what follows.…”

“…One natural application of the above result is to the problem of subregion duality. By studying the subregion −L ≤ x ≤ L the paper [39] showed that it is possible to reconstruct the bulk region given by X 2 + Z 2 ≤ L 2 . The curve X 2 + Z 2 = L 2 is the geodesic in the AdS 4 spacetime that connects the endpoints X = ±L, Z = 0 on this constant X + slice, so it is the Ryu-Takayanagi surface [40] in lightcone quantization.…”

Gravitational dynamics from collective field theory

“…Collective field theory was first used to explore this duality in [34]. It was then further developed in a series of articles [35][36][37][38][39][40][41][42][43][44] to achieve a detailed holographic mapping from the 2 + 1 dimensional vector model CFT to the higher spin gravity in AdS 4 . This mapping is constructed at large N so that we obtain a free theory in the bulk of AdS 4 .…”

Bilocal holography and locality in the bulk

“…This change of coordinates is highly non-trivial and the correctness of this step was verified by showing that the complete set of generators of the conformal field theory are mapped into those of the higher spin gravity [19]. Bilocal holopgraphy has been developed in a number of interesting directions [20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”